ACTUARIAL MODELLING OF EXTREMAL EVENTS USING TRANSFORMED GENERALIZED EXTREME VALUE DISTRIBUTIONS AND GENERALIZED PARETO DISTRIBUTIONS A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Zhongxian Han, B.S., M.A.S. ***** The Ohio State University 2003 Doctoral Examination Committee: Bostwick Wyman, Advisor Approved by Peter March Gerald Edgar Advisor Department of Mathematics ABSTRACT In 1928, Extreme Value Theory (EVT) originated in work of Fisher and Tippett de- scribing the behavior of maximum of independent and identically distributed random variables. Various applications have been implemented successfully in many fields such as: actuarial science, hydrology, climatology, engineering, and economics and finance. This paper begins with introducing examples that extreme value theory comes to encounter. Then classical results from EVT are reviewed and the current research approaches are introduced. In particular, statistical methods are emphasized in detail for the modeling of extremal events. A case study of hurricane damages over the last century is presented using the \excess over threshold" (EOT) method. In most actual cases, the range of the data collected is finite with an upper bound while the fitted Generalized Extreme Value (GEV) and Generalized Pareto (GPD) distributions have infinite tails. Traditionally this is treated as trivial based on the assumption that the upper bound is so large that no significant result is affected when it is replaced by infinity. However, in certain circumstances, the models can be im- proved by implementing more specific techniques. Different transforms are introduced to rescale the GEV and GPD distributions so that they have finite supports. ii All classical methods can be applied directly to transformed models if the upper bound is known. In case the upper bound is unknown, we set up models with one additional parameter based on transformed distributions. Properties of the transform functions are studied and applied to find the cumulative density functions (cdfs) and probability density functions (pdfs) of the transformed distributions. We characterize the transformed distribution from the plots of their cdfs and mean residual life. Then we apply our findings to determine which transformed distribution should be used in the models. At the end some results of parameter estimation are obtained through the maximum likelihood method. iii Dedicated to My Parents iv ACKNOWLEDGMENTS I am deeply indebted to my advisor Dr. Wyman, for his general support and his patience over these past years. Particularly he always gives me directions for searching a field which I am really interested in. I thank Dr. Lijia Guo from University of Central Florida. She has my gratitude both for introducing me the topic of this thesis and directing me for supporting materials. My discusss with her were very helpful in deciding the final topic. I thank the members of my Examination Committee (Dr. Bostwick Wyman, Peter March, and Gerald Edgar of the Department of Mathematics and Dr. Mario Peruggia of the Department of Statistics) for their comments, criticisms and suggestions. Also, I thank my girlfirend, Yun Fu for being a good listener. Also her encour- agement and enlightment helped me to finish this paper more easily. This document was prepared using -LATEX, the American Mathematical So- AMS ciety's LATEXmacro system. Most of the results involving statistics run under S-plus and many function plots are generated by Matlab. v VITA December 18, 1975 . Born - Lanzhou, P.R.China. 1997 . B.S., Peking University Beijing, China. 1997 - present . Graduate Teaching and Research Asso- ciate, The Ohio State University. 2001 . Master in Applied Statistics The Ohio State University. FIELDS OF STUDY Major field: Mathematics Specialization: Extreme Value Theory vi TABLE OF CONTENTS Abstract . ii Dedication . iv Acknowledgments . v Vita . vi List of Tables . ix List of Figures . x CHAPTER PAGE 1 Introduction of Extreme Value Theory . 1 1.1 Extremal Events . 1 1.2 Modeling Extremal Events . 5 1.3 Extreme Value Theory . 9 1.4 Generalized Extreme Value Distributions (GEV) . 13 1.5 Generalized Pareto Distributions (GPD) . 17 2 Statistical Methods in Application of EVT . 21 2.1 Exploratory Data Analysis . 21 2.2 Parameter Estimation for the GEV . 26 2.3 The Excess Over Threshold Method . 29 2.4 Approximate Normality of the MLE and Profile Likelihood 33 3 Case Study: Hurricane Damages over the Last Century . 39 3.1 Data Resource and Information . 39 3.2 Fitting Data by the EOT Method: Threshold Selection . 41 3.3 Statistical Analysis of S-Plus Plots . 43 vii 4 Transforms of GEV and GPD . 47 4.1 Class of Transforms . 47 4.2 Properties of Transformed GEV and GPD Distributions . 54 4.3 Conditional Expectation of Excess Over Threshold . 62 5 Modeling over Transformed Distributions . 68 5.1 Model Selection: Transformed or Not? . 68 5.2 Parameter Estimations of Transformed Distributions . 73 5.3 Future Research Directions . 77 Bibliography . 80 viii LIST OF TABLES TABLE PAGE 1.1 10 Largest Catastrophe Events in US (Source: ISO's PCS) . 3 1.2 California Earthquake Data (1971 { 1994) . 4 1.3 Analog between CLT and EVT . 11 3.1 Some descriptive statistics of hurricane data . 39 4.1 cdfs and pdfs of transformed GEV distributions . 61 4.2 cdfs and pdfs of transformed GPD distributions . 61 5.1 Maximum points of the mean excess function for ξ = 0:1; 0:2; :::; 2:0 . 72 ix LIST OF FIGURES FIGURE PAGE 1.1 Density Plots of Generalized Extreme Distributions . 14 3.1 Scatter plot of the hurricane data . 40 3.2 Mean residual plot of hurricane data . 41 3.3 Parameter estimations for different thresholds . 42 3.4 Goodness-of-fit Plots for hurricane data, u = 500 . 43 3.5 Profile log-likelihood estimate of shape parameter ξ . 44 3.6 Profile log-likelihood estimate of 10 year return level . 46 1 4.1 pdf of hyperbolic transformed GPD family when ξ = ; 1; 2 . 58 2 1 4.2 pdf of logarithmic transformed GPD family when ξ = ; 1; 2 . 59 2 4.3 pdf of logarithmic transformed GPD family whenu ~ = 10; 20; 30 . 60 4.4 Plot of m(λ)............................... 64 4.5 Plot of mr(λ) .............................. 66 5.1 Mean residual life plot for daily rainfall data, Source: Coles[6] . 70 5.2 Mean residual life plot for fire data, Source: Embretchs[9] . 71 5.3 Mean excess function of transformed GPD for ξ = 0:1; 0:2; :::; 2:0 . 72 x CHAPTER 1 INTRODUCTION OF EXTREME VALUE THEORY 1.1 Extremal Events Extremal events are also called \rare" events. Extremal events share three charac- teristics: \relatively rareness," \huge impact," and \statistical unexpectness." From the name of it, extremal events are extreme cases; that is, the chance of occurrence is very small. But over the last decades, we have observed several. We list some recent events with a large impact in time order: Hurricane Andrew (1992), Northridge earthquake (1994), terrorism attack (2001), disintegration of space shuttle Columbia (2003), winter blizzard of the northeast (2003). If we assume those events are in different fields and independent of each other, the probability of the occurrence of each event is small. The terminology t-year event has been used by hydrologists to describe how frequent a certain event will occur. In general, suppose for every year, the probability of a certain event is p, where p is relatively small. Given that the events of each year are independent of each other, the number of the year when the event occurs follows a geometric distribution with parameter p, and thus has an 1 expectation of . On the other words, we say Hurricane Andrew is a 30-year event p 1 means that for every year the probability of having a hurricane which is more severe 1 than Hurricane Andrew is approximately . 30 The enormous impact of catastrophic events on our society is deep and long. Not only we need to investigate the causes of such events and develop plans to protect against them, but also we will have to resolve the resulting huge financial loss. Finan- cial plans have been established for the reduction of the impact of extremal events. For example, the Homeland Security Act of 2002 was passed on Nov 15, 2002 to prevent future terrorism attacks. In the insurance industry of, the magnitude of property- casualty risk has become a major topic of research and discussion. Traditionally, insurers buy reinsurance to hedge against catastrophic risk. For a catastrophic event causing more than $5 billion of insured losses, the overwhelming majority is not cov- ered by reinsurance, according to Froot and Connell [11]. On December 11, 1992, the Chicago Board of Trade launched the first generation of catastrophe insurance futures, also called CAT-Futures. The related future PCS options were launched shortly after. CAT-Futures provides a way of transferring insurance risk to the finan- cial market and CAT-Futures can be traded as standardized securities. Table 1.1 lists 10 catastrophic events that caused the largest losses in history of the United States. Since the population and fortune grow at a relatively constant rate, we expect larger catastrophic events entering the list in the future. The task of understanding and resolving the underlying risk is still a big challenge for researchers. 2 Year Catastrophic Event Estimated Losses (in billions) 2001 Fire and explosion 20.3 1992 Hurricane Andrew 15.5 1994 Northridge earthquake 12.5 1989 Hurricane Hugo 4.2 1998 Hurricane Georges 3.0 2001 Tropical Storm Allison 2.5 1995 Hurricane Opal 2.1 1999 Hurricane Floyd 2.0 2001 St.Louis hailstorm 1.9 1993 Midwest blizzard 1.8 Table 1.1 10 Largest Catastrophe Events in US (Source: ISO's PCS).